Jensen and Hermite-Hadamard type inclusions for harmonical h -Godunova-Levin functions

: The role of integral inequalities can be seen in both applied and theoretical mathematics ﬁelds. According to the deﬁnition of convexity, it is possible to relate both concepts of convexity and integral inequality. Furthermore, convexity plays a key role in the topic of inclusions as a result of its deﬁnitional behavior. The importance and superior applications of convex functions are well known, particularly in the areas of integration, variational inequality


Introduction
Since we know that interval analysis has a veritably broad history, Moore [1] first developed the interval and interval-valued functions in his work in the 1950s. This research field has attracted the attention of the mathematical community since it was established. There are many applications of interval analysis in global optimization algorithms and constraint solving algorithms. In contrast, calculation error has long been a problematic issue in numerical analysis. The accumulation of calculation errors can make the calculation result meaningless, so interval analysis has attracted much attention as a tool to solve uncertainty problems [2,3]. For the last five decades, It has been used in a variety of fields, including neural network output optimization [4], computer graphic [5], interval differential equation [6], aeroelasticity [7] and so on. The fusion of integral inequalities with intervalvalued functions (IVFS) has resulted in many insightful findings in recent decades. Since the invention of interval analysis, researchers studying inequalities have been interested in seeing if the inequalities found in the below results can be substituted with inclusions. Among these are Beckenbach type inequalities and Minkowski types for IVFS developed by Roman-Flores [8]. Moreover, Costa [9] established Opial-type inclusion for IVFS.
Classical Hermite-Hadamard inequality (H-H) is as follows: (1.1) Due to its geometrical interpretation, the H-H inequality is considered one of the classics of elementary mathematics. In addition to being generalized and refined, the function is now extended to cover various classes of convexity. Many inequalities have been revealed by the convexity of functions over time in mathematics and other scientific fields, including economics, probability theory, and optimal control theory, as well as in economics. In probability theory, a convex function applied to the expected value of a random variable is always bound by the expected value of its convex function. Further, Jensen's inequality is a probabilistic inequality, and its beauty lies in the fact that several well-known inequalities can be deduced from it, including the arithmetic-geometric mean inequality and Holder's inequality based on the expected values for convex and concave transforms of random variables. For different extensions and conceptions of these inequalities [10][11][12][13][14][15][16][17][18][19]. Initially, Işcan present the concept of harmonical convexity in 2014 and created various H-H inequalities for this form of convexity [20]. In the case of harmonical convex functions, some refinements of such inequalities have been investigated [21][22][23][24][25][26].
Noor et al. [27] established harmonical h-convex functions and developed a revised form of H-H inequalities in 2015. In addition to interval analysis, Dafang et al. extended H-H and Jensen type inequalities to h-convex and harmonic h-convex in the context of IVFS [28,29]. We refer interested readers to some new research on harmonical h-convexity [30][31][32][33][34][35]. Based on the notion of the h-GL function, Kilicman et al. developed the following inequality [36]. As a step forward, Afzal et al. developed these inequalities in 2022 for the generalized class of h-Godunova-Levin functions and (h 1 , h 2 )-Godunova-Levin functions in the context of interval-valued functions using inclusion relation [37,38]. The beauty of this class of convexity is that inequality terms are straightforward to deduce and generalize. Moreover, Baloch et al. developed the Jensen-type inequality for harmonic h-convex functions [39].
Inspired by [29,[37][38][39], we present harmonical h-Godunova-Levin functions as a new class of convexity based on inclusion relation for IVFS. As part of our analysis, we first derived new variants of the H-H inequality, and then we used this new class to represent the Jensen inequality. Additionally, we provide several examples to illustrate how our key findings can be applied.
Finally, the rest of the paper is organized as follows. In Section 2, preliminary information is provided. The key conclusions are described in Section 3. Section 4 contains the conclusion.

Preliminaries
For the notions which are used in this paper and are not defined here, we refer [28]. Let's say I represent a set of real numbers in the form of a pack of all intervals of R, [s] ∈ I is defined as where real interval [s] is compact subset of R. There is a degeneration of the interval [s] when s = s. In this case, we are denoting the bundle of all intervals in R by R I and use R I + for the collection of all positive intervals. The inclusion "⊆" is established as For any arbitrary κ ∈ R and [s], the κ[s] is defined as for all s, r ∈ S and κ ∈ [0, 1].

Main results
In this section firstly we define a novel class of convexity called harmonic h-GL IVFS.
for all s, r ∈ S and κ ∈ (0, 1). If the inclusion is change from ⊆ to ⊇ in Definition 3.1, then f is called harmonical h-GL concave IVF. Harmonical h-GL convex and concave IVFS are represented by S GHX 1 h , S , R I + and S GHV 1 h , S , R I + , respectively.
It shows that f ∈ S GHX The proof similar to Proposition 3.1.
Then 1 Multiplying both sides by h 1 2 , we have The above inequality is integrated over (0, 1), we have It follows that Similarly, This implies that Divide both sides by 1 2 first inclusion of (3.3) is proved, According to our hypothesis, Adding above two inclusions and integrate over (0, 1), we have Since at x = 1 2 both integrals are equal, which implies that Dividing by 2, we obtain the desired result, (3.7) By combining (3.6) and (3.7), we obtain the desired result This completes the proof.
If h(x) = 1 x , then Theorem 3.1 reduces to harmonical convex IVFS: If h(x) = 1 x s , then Theorem 3.1 reduces to harmonical s-IVFS: Thus, we obtain which demonstrates the result described in Theorem 3.1. , Proof On integration over (0, 1), we have Then, above inequality become as Similarly for interval 2sr s+r , r , we have Adding above inclusions (3.8) and (3.9), we have This completes the proof.
On integration over (0, 1), we have It follows that The proof is completed. Then, , and N(s, r) Proof. According to our hypothesis, we have N(s, r) .
On integration over (0, 1), we have Therefore, the proof is completed.